Conditional expectation and probability

Conditional expectation

D4.1.1 Let `(Omega, ccF, P)` be a ps, and `ccG sub ccF` a sub-`sigma`-field. Let `X` be a random variable with `E|X| < oo`, then the conditional expectation of `X` given `ccG`, `E[X|ccG] : Omega -> RR` and:

`E(X|ccG)` is `ccG`-measurable
`int_A E(X|ccG) dP = int_A X dP quad AA A in ccG`

Remarks:

`Omega in ccG` so `E(E(X|ccG)) = E(X)`
any `ccG` measurable function which differs from `E(X|ccG)` on a set of measure 0 also qualifies as a conditional expection of `X`

P4.1.1 For any rv `X` with `E|X| < oo` and a sub-`sigma`-field `ccG`, `E(X|ccG)` exists and is unique wp1.

P4.1.2 If `X` an rv `E|X| < oo`, `ccG` a sub `sigma`-field.
If `Z` is an integrable `ccG`-measurable rv st for a `pi`-class `D` with `sigma(D) = ccG` `EX = EZ` and `int_A Z dP = int_A X dP quad AA A in D` then `Z = E(X|ccG) "wp1"`.

C4.1.3 Let `ccG_1, ccG_2` be sub `sigma`-fields of `ccF` and `X, Y` be integrable rvs.

if `sigma(X)` and `ccG_1` are indepdendent, then `E(X|G_1) = E(X) "wp1"`
if `int_(A_1 nn A_2) X dP = int_(A_1 nn A_2) Y dP quad AA A_i sub ccG_i`, then `E(X|sigma(ccG_1 uu ccG_2)) = E(Y | sigma(ccG_1 uu ccG_2))`

P4.1.4 (Properties of CE). Let `X, Y` be random variables on `(Omega, ccF, P)`, `ccG sub ccF` a sub-`sigma`-field. Then:

`E(1 | ccG) = 1`
`E(X | ccG) >= 0 "wp1"` if `X >= 0 "wp1"`
`E(cX | ccG) = c E(X|ccG) "wp1"` if `|EX| <= oo`, `c in RR`
`E(X + Y | ccG) = E(X | ccG) + E(Y | ccG) "wp1"` if `E|X+Y| <= oo`
`E(X|ccG) = X "wp1"` if `X` is `ccG`-measurable
If `ccG_1 sub ccG_2 sub F`, `|EX| <= oo` then `E(E(X|G_2)|G_1) = E(X|G_1) = E(E(X|G_1)|G_2)`

T4.1.6 `{X_n}`, Y be random variables with `E|Y| < oo` and `ccG sub ccF` on `(Omega, ccF, P)` then:

(MCT) If `Y <= X_n uarr "wp1"` then `E(X_n | ccG) -> E(X | ccG) "wp1"`
(Fatou) If `Y <= X_n quad AA n >=1 "wp1"` then `E(ul lim X_n | ccG) <= ul lim E(X_n | ccG) "wp1"`
(DCT) If `X_n ->^"wp1" X` and `|X_n| <= |Y| "wp1" quad AA n >= 1` then `E(X_n | ccG) -> E(X | ccG) "wp1"`

T4.1.7 (Jensen). `Y` a `ccF`-measurable with `|EY| <= oo` and `g` be a finite convex function on `RR` with `|Eg(Y)| <= oo`. If for some `sigma`-field `ccG`:

`X = E(Y | ccG) "wp1"` OR
`X` is `ccF`-measurable with `X <= E(X|ccG) "wp1"` and `g uarr`

then `g(X) <= E(g(X) | ccG "wp1"`

D4.1.2 If `EY^2 < oo`, the conditional variance of `Y` given sub-`sigma`-field `ccG` `Var(Y | ccG) = E(Y^2 | ccG) + E^2(Y | ccG)`

T4.1.8 Let `EY^2 < oo` then `Var(Y) = Var(E(Y| ccG)) + E(Var(Y|ccG))`

Conditional probability

D4.2.1 Let `(Omega, ccF, P)` be a ps. For a `B in ccF` and sub `sigma`-field `ccG sub ccF`, the conditional probability of `B` given `ccG`, `P(B | ccG) = E(I_B | ccG)`.

Remarks:

conditional probability is a just a conditioanl expection of an indicator rv, satisfying:
- `P(B | ccG)` is `ccG`-measurable
- `int_A P(B | ccG) dP = int_A I_B dP = int I_(A nn B) dP quad AA A in ccG`, ie. `P(A nn B) = E( P(B|ccG)I_A)`

The properties of conditional expectation lead to the following properties of conditional probability:

`0 <= P(A | ccG) <= 1 "wp1"`
`P(A|ccG) = 0 "wp1" iff P(A)=0`
`P(A|ccG) = 1 "wp1" iff P(A)=1`
If `{A_n} sub ccF` are disjoint sets, then `P(UU_i A_i | ccG) = sum_i P(A_i | ccG) "wp1"`
If `A_n in F quad n >= 1` and `lim_(n->oo) A_n = A` then `lim_(n->oo) P(A_n | ccG) = P(A | ccG) "wp1"`

Above suggests that `mu_(ccG)(*) = P(* | ccG)` is sub-additive wp1 for a given set of `{A_i}`. However, this may not be subadditive in general wp1, as the probability 1 set may change from set to set. Hence we may not be able to find a common probability 1 set `=> P(* | ccG)` may not be a pm on `ccF`.

D4.2.2 Let `ccF_1, ccG` be sub-`sigma`-fields of events in `(Omega, ccF, P)`. A regular conidtional probability on `F_1|ccG` is a function `mu: ccF_1 xx Omega -> [0,1]` satisfying:

`AA B in ccF_1`, `mu(B, omega) = P(B | ccG)(omega)`
`AA omega in Omega`, `mu(*, omega)` is a pm on `ccF_1`

T4.2.1 Let `P_omega(A) := P(A, w)` be a regular conditional probability. Given `ccG`, `X` is `ccF`-measuralbe with `|EX| <= oo` then `E(X|ccG)(omega) = int_Omega X dP_omega`